Sold in part to Benefit the John C.M. Nash Trust
Post Lot Text
GOLD MEDALS FOR GAMES
Game theory is the science of strategy. In most political, social, and economic interactions, the choices of the participants impact one another. Therefore each has to think about others’ decisions, recognizing that conversely the others are thinking about theirs. Amazingly, this complex thinking about thinking can be systematized. Research on how to do so has led to several Nobel prizes, starting with those to John Nash, Reinhard Selten, and John Harsanyi 25 years ago in 1994.
People have been making strategic decisions for millennia, perhaps most dramatically in military conflicts, but also in dealing with family, friends, business rivals, and sports opponents. They have developed a lot of art specific to their experiences. Game theory builds on that, identifies common features, and develops general principles which can then facilitate strategic thinking for new applications.
The first such successful theorizing came from the polymath John von Neumann and his economist coauthor Oskar Morgenstern. They considered two-person games of pure conflict (win-lose or zero-sum games) where one player’s gain necessarily entails the other’s loss. Sports contests are the best-known examples. Von Neumann and Morgenstern proved a mathematical theorem: that zero-sum games always have an equilibrium, a pair of strategies, one for each player, such that neither can gain by deviating to a different strategy. They also showed that the equilibrium often involves mixing moves: acting randomly on any one occasion so as to keep the other player guessing. For example, a penalty kicker in soccer should not always kick to the goalie’s left. The theorem also shows how to calculate the proportions in which the kicker should mix left and right kicks.
But most games have more than two players, and can be win-win (international trade) or lose-lose (nuclear arms race). Most games are not one-offs; they can be repeated, or be a part of a sequence of different interactions. Advances in game theory have enabled us to understand such games and their outcomes.
John Nash’s first Nobel-worthy contribution proved that many-player, non-zero-sum games also have an equilibrium: a configuration of all players’ strategies where no one has any incentive to switch to a different strategy. With Nash equilibrium in hand as their “solution concept” for understanding and predicting strategic behavior, social scientists could now investigate countless applications that had previously eluded systematic investigation, in fields from economics and law to political science and military strategy, even biology.
As fellow Nobel Laureate Roger Myerson put it in his own homage: “Nash carried social science into a new world where a unified analytical structure can be found for studying all situations of conflict and cooperation…The formulation of Nash equilibrium has had a fundamental and pervasive impact in economics and the social sciences, comparable to that of the discovery of the DNA double helix in the biological sciences.”
The best-known example of Nash equilibrium in action is the Prisoners’ Dilemma. Nash’s Ph.D. adviser Albert Tucker invented the story of this game. The police interrogate two suspects separately, and suggest to each that he or she should fink on the other and turn state’s evidence. “If the other does not fink, then you can cut a good deal for yourself by giving evidence against the other; if the other finks and you hold out, the court will treat you especially harshly. Thus no matter what the other does, it is better for you to fink than not to fink—finking is your uniformly best or ‘dominant’ strategy.” This is the case whether the two are actually guilty, as in some episodes of Law and Order, or innocent, as in LA Confidential. Of course, when both fink, both fare worse than if both had held out. Although holding out is jointly better, it does not survive their separate temptations to fink—it is not a Nash equilibrium!
These dilemmas arise everywhere. Perhaps the biggest concerns action to combat climate change. Each country wants to continue its economic growth, but more economic activity usually requires more greenhouse gas emissions. Any one country’s emissions add only a little to the total accumulation of these gases. But when all countries give in to the temptation to pursue growth, the effect is substantial and the resulting warming puts humanity’s future at risk.
Such dilemmas can be resolved in multiple ways. First, if players interact repeatedly over a long time horizon, then fear of lost future cooperation may keep them cooperating today; this is the well-known tit-for-tat strategy. Second, a “large” player who suffers disproportionately more from complete finking may act cooperatively, tolerating small-fry finking. Thus the United States bears a disproportionate share of the costs of its military alliances. Finally, if the group as a whole will do better in its external relations if it enjoys internal cooperation, then biological instincts or social norms that support cooperation and punish cheating can arise and prevail. Many biologists claim that the evolution of cooperation in humans has such game-theoretic origin. In fact, any “evolutionarily stable” outcome of an evolving biological system corresponds to a Nash equilibrium of an “evolutionary game.”
Nash’s formulation of the concept of “Nash equilibrium” and proof of equilibrium existence would have been enough. But he had a second contribution worthy in its own right of a Nobel prize: a theory of bargaining known as “Nash bargaining” that is the mainstay of most subsequent theories and applications to labor-management negotiations, international trade, and merger analysis, among other areas.
Unlike Nash, whose star immediately shone, Reinhard Selten’s seminal work languished nearly unknown at first. Indeed, it was not until 1980 that Selten’s earliest Nobel-cited work (published in 1965 in Staatswissenschaft) was mentioned in an English-language review. But then the floodgates opened dramatically. In 1982, five brilliant young theorists—Ariel Rubenstein on bargaining; David Kreps and Robert Wilson on reputation; and Paul Milgrom and John Roberts on entry deterrence—all built on Selten’s foundation with seminal contributions that may, one day soon, win their own Nobel prizes.
Selten’s contributions, most notably, “subgame perfection,” refine Nash’s concept: in some situations, many sorts of outcomes can be consistent with equilibrium, making it difficult to gain insight into the economic phenomena being studied. However, Selten showed that Nash equilibria can be refined down to a smaller set, by discarding information that fails to pass certain plausibility tests (e.g., by disallowing Nash equilibria in which any player makes a non-credible threat which, if put to the test, she would back away from carrying out).
Together, John Nash and Reinhard Selten laid the groundwork for a blossoming of the social sciences that continues to this day. The discoveries honored by these Nobel Prizes changed the game of how we understand and compete in our ever-changing world. We hope that whoever secures these treasures finds a way to share them with the world, to inform and inspire future generations.
- Avinash Dixit, David McAdams, and Susan Skeath
Avinash Dixit is University Professor of Economics, Emeritus, at Princeton and the author of Art of Strategy (with Barry Nalebuff; W.W. Norton, 2008). David McAdams is Professor of Economics at Duke University and the author of Game-Changer (W.W. Norton, 2014). Susan Skeath is Professor of Economics at Wellesley College and Acting Director of the Quantitative Reasoning Program. Together they are coauthors of the forthcoming fifth edition of Games of Strategy (W.W. Norton, 2020).